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Overlay Stitch Meshing

Overlay Stitch Meshing. A competitive algorithm for no-large-angle triangulation. Don Sheehy Joint work with Gary Miller and Todd Phillips To appear at ICALP 2007. The Problem. Input: A Planar Straight Line Graph. The Problem. Input: A Planar Straight Line Graph.

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Overlay Stitch Meshing

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  1. Overlay Stitch Meshing Don Sheehy Overlay Stitch Meshing

  2. A competitive algorithm for no-large-angle triangulation Don Sheehy Joint work with Gary Miller and Todd Phillips To appear at ICALP 2007 Don Sheehy Overlay Stitch Meshing

  3. The Problem Input: A Planar Straight Line Graph Don Sheehy Overlay Stitch Meshing

  4. The Problem Input: A Planar Straight Line Graph Output: A Conforming Triangulation Don Sheehy Overlay Stitch Meshing

  5. The Problem Input: A Planar Straight Line Graph Output: A Conforming Triangulation Don Sheehy Overlay Stitch Meshing

  6. Why would you want to do that? Don Sheehy Overlay Stitch Meshing

  7. Don Sheehy Overlay Stitch Meshing

  8. Don Sheehy Overlay Stitch Meshing

  9. Don Sheehy Overlay Stitch Meshing

  10. Don Sheehy Overlay Stitch Meshing

  11. Don Sheehy Overlay Stitch Meshing

  12. What went wrong? Don Sheehy Overlay Stitch Meshing

  13. ? ? What if you don’t know the function? Don Sheehy Overlay Stitch Meshing

  14. 2 Definitions of Quality 1. No Large Angles [Babuska, Aziz 1976] Don Sheehy Overlay Stitch Meshing

  15. 2 Definitions of Quality 1. No Large Angles [Babuska, Aziz 1976] 2. No Small Angles Don Sheehy Overlay Stitch Meshing

  16. No Small Angles Don Sheehy Overlay Stitch Meshing

  17. No Small Angles You may have heard of these before. Delaunay Refinement Sparse Voronoi Refinement Quadtree Don Sheehy Overlay Stitch Meshing

  18. Paying for the spread Spread = L/s L s Don Sheehy Overlay Stitch Meshing

  19. Paying for the spread Optimal No-Large-Angle Triangulation Don Sheehy Overlay Stitch Meshing

  20. Paying for the spread What if we don’t allow small angles? Don Sheehy Overlay Stitch Meshing

  21. Paying for the spread What if we don’t allow small angles? O(L/s) triangles! Don Sheehy Overlay Stitch Meshing

  22. Paying for the spread What if we don’t allow small angles? Fact: For inputs with NO edges, no-small-angle meshing algorithms produce output with O(n log L/s) size and angles between 30o and 120o O(L/s) triangles! Don Sheehy Overlay Stitch Meshing

  23. What to do? Small input angles can force even smaller ouput angles. [Shewchuk ’02] Don Sheehy Overlay Stitch Meshing

  24. No Large Angles Don Sheehy Overlay Stitch Meshing

  25. Polygons with Holes [Bern, Mitchell, Ruppert 95] – - All triangles are nonobtuse. - Output has O(n) triangles. Don Sheehy Overlay Stitch Meshing

  26. Polygons with Holes [Bern, Mitchell, Ruppert 95] – - All triangles are nonobtuse. - Output has O(n) triangles. Does not work for arbitrary PSLGs Don Sheehy Overlay Stitch Meshing

  27. Paterson’s Example Requires (n2) points. O(n) points O(n) lines Don Sheehy Overlay Stitch Meshing

  28. Paterson’s Example Requires (n2) points. O(n) points O(n) lines Don Sheehy Overlay Stitch Meshing

  29. Paterson’s Example Requires (n2) points. O(n) points O(n) lines Don Sheehy Overlay Stitch Meshing

  30. Paterson’s Example Requires (n2) points. O(n) points O(n) lines Don Sheehy Overlay Stitch Meshing

  31. Paterson’s Example Requires (n2) points. O(n) points O(n) lines Don Sheehy Overlay Stitch Meshing

  32. Paterson’s Example Requires (n2) points. O(n) points O(n) lines Don Sheehy Overlay Stitch Meshing

  33. Propagating Paths Don Sheehy Overlay Stitch Meshing

  34. Propagating Paths Don Sheehy Overlay Stitch Meshing

  35. Propagating Paths First introduced by Mitchell [93] Later Improved by Tan [96] Worst Case Optimal Size O(n2) Angle bounds: 132o Don Sheehy Overlay Stitch Meshing

  36. The OSM Algorithm(Overlay Stitch Meshing) Don Sheehy Overlay Stitch Meshing

  37. The OSM Algorithm(Overlay Stitch Meshing) • Results: • Angles bounded by 170o • O(log L/s) competitive size Don Sheehy Overlay Stitch Meshing

  38. The OSM Algorithm(Overlay Stitch Meshing) Don Sheehy Overlay Stitch Meshing

  39. The OSM Algorithm(Overlay Stitch Meshing) Don Sheehy Overlay Stitch Meshing

  40. The OSM Algorithm(Overlay Stitch Meshing) Don Sheehy Overlay Stitch Meshing

  41. The OSM Algorithm(Overlay Stitch Meshing) Don Sheehy Overlay Stitch Meshing

  42. The OSM Algorithm(Overlay Stitch Meshing) Don Sheehy Overlay Stitch Meshing

  43. The OSM Algorithm(Overlay Stitch Meshing) Don Sheehy Overlay Stitch Meshing

  44. The OSM Algorithm(Overlay Stitch Meshing) Don Sheehy Overlay Stitch Meshing

  45. The OSM Algorithm(Overlay Stitch Meshing) Don Sheehy Overlay Stitch Meshing

  46. The OSM Algorithm(Overlay Stitch Meshing) Don Sheehy Overlay Stitch Meshing

  47. The OSM Algorithm(Overlay Stitch Meshing) Don Sheehy Overlay Stitch Meshing

  48. The OSM Algorithm(Overlay Stitch Meshing) Don Sheehy Overlay Stitch Meshing

  49. The OSM Algorithm(Overlay Stitch Meshing) • An Overlay Edge is kept if 1. It does not intersect the input, OR 2. It forms a good intersection with the input. at least 30o Don Sheehy Overlay Stitch Meshing

  50. Angle Guarantees Don Sheehy Overlay Stitch Meshing

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